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We deal with the $(\mathbb{Z}\omega, \chi)$-theta Fock-Bargmann spaceconsisting of holomorphic automorphic functions associated to given discretesubgroup in $\mathbb{C}^g$ of rank one and given character $\chi$. We giveconcrete description of its elements, it looks like a tensor product of a thetaFock-Bargmann space on the complex plane $\mathbb{C}$ and the classicalFock-Bargmann space on $\mathbb{C}^{g-1}$. Moreover, we construct anorthonormal basis and we give explicit expression of its reproducing kernelfunction in terms of Riemann theta function.

Concrete description for rank one $(\Gamma, \chi)$-theta Fock-Bargmann space in high dimension